The cover's a bit awry when the paper's torn (7)

I believe the answer is:

wrapper

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'bit awry when the paper's torn' is the definition.
I can't tell whether this definition defines the answer.

'the cover's a' is the wordplay.
'the cover' becomes 'wrap' (I've seen this before).
'a' becomes 'per' (eg 'one a day' means 'one per day').
'wrap'+'per'='WRAPPER'

Can you help me to learn more?

(Other definitions for wrapper that I've seen before include "Covering for a sweet" , "band" , "Hershey Bar comes included in this" , "cover version" , "Packaging" .)

I've seen this clue in the Evening Standard.

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